3.1 Model development
3.1.1 GBM
To further clarify the formation mechanism of rock failure characteristics under particle impact and analyze the influence mechanism of confining pressure on crack propagation, the discrete element method is used to carry out numerical simulation. The rock consists of several tiny particles with different contact models between particles in the PFC simulation analysis. The parallel-bonded model (PBM) adds a bonding function to the linear contact model, allowing the contact model to transfer forces and moments. PBM is commonly used in rock, concrete, and other materials. However, the uniaxial test simulation using the micromechanical parameters of the PBM obtained by inversion often has a high tensile strength. The GBM proposed by Potyondy combines the PBM with the smooth joint model (SJM) to establish a complex rock model with polygonal grains bonded to each other 23. The tension–compression ratio of the GBM is in the range of brittle rocks. The model can reflect the mechanical behaviors of the rock crystal friction and cementation failure. This study is based on the GBM.
The setting of the GBM for the rock is shown in Fig. 11. Particles are formed in a quadrilateral area, and then Thiessen polygons are superimposed on the particles to group the particles in each crystal. A granite with the GBM is constructed by assigning the PBM to the particles inside the crystal and SJM to the particles between the crystal boundaries. Figure 12 shows the failure modes of the two mechanical models in the GBM. The bond breaks when the maximum normal stress or shear stress between particles exceeds the bond strength of the PBM, and the forces, moments, and stiffness associated with the bond are removed. When the strength exceeds the tolerance limit of the SJM, the particles slide along the edge interface.
3.1.2 Model calibration
There is no distinct quantitative relationship between the microscopic parameters of the discrete-element model and macroscopic mechanical parameters of rock mass obtained by laboratory tests. Therefore, to obtain reasonable microscopic parameters, different macroscopic mechanical parameters need to be selected for the simulation trial. The simulation results obtained by the trial calculation are compared to laboratory test results. Until the macroscopic mechanical parameters and failure modes of the simulation results are similar to the laboratory test results, the mesoscopic parameters used in the trial calculation can be considered reasonable 24–28. This study is based on uniaxial compression and tensile experiments to calibrate the macroscopic mechanical parameters of the particles. The specimen dimensions used in the numerical simulation are Φ50 mm × 25 mm, while the loading speed is 0.001 m/s to ensure that the model specimen is in a quasi-static equilibrium. The calibration results are shown in Table 2. The uniaxial compression numerical simulation stress–strain curve is shown in Fig. 13.
Table 2
Microscopic parameter calibration results of the granite discrete-element model
Basic parameters of rock particles |
Minimum particle radius (mm) | 0.13 |
Ratio of maximum to minimum disk radius | 1.66 |
Particle density (kg⋅m− 3) | 2410 |
Contact modulus (GPa) | 14.2 |
Porosity | 0.08 |
Normal to shear stiffness ratio | 1.5 |
Intragranular (PBM) |
Parallel bond modulus (GPa) | 14.2 |
Parallel bond normal to shear stiffness ratio | 1.5 |
Parallel bond normal (tensile) strength (MPa) | 50 |
Parallel bond cohesion (MPa) | 100 |
Parallel bond friction angle (°) | 30 |
Intergranular (SJM) |
Smooth-joint normal stiffness factor (N⋅m− 1) | 8.1 × 1014 |
Smooth-joint shear stiffness factor (N⋅m− 1) | 3.8 × 1014 |
Smooth-joint bond normal (tensile) strength (MPa) | 18 |
Smooth-joint bond cohesion (MPa) | 180 |
Smooth-joint bond friction angle (°) | 20 |
The physical and mechanical parameters of the calibrated rock specimen model and actual rock specimen are shown in Table 3.
Table 3
Macro mechanical parameters of the rock sample and discrete-element model
Structure | Density (kg⋅m− 3) | Elastic modulus (GPa) | Compressive strength (MPa) | Poisson ratio |
Rock sample | 2410 | 27.8 | 156.8 | 0.24 |
Discrete-element rock model | 2410 | 29.3 | 157.0 | 0.24 |
3.1.3 Model of particle-impact rock breaking
In the process of rock excavation, the rock in a loaded state is broken by the impact of particles at certain speed, which reduces the strength of the rock. Considering the computational efficiency, the particle-impact rock process is simplified to a two-dimensional numerical model. Figure 14 shows the particle-impact rock model established by PFC2D. In the discrete element numerical simulation, the particles are constrained by frictionless rigid walls on both sides and at the base, and the confining pressure is loaded on the rock using the walls on both sides. Considering the calculated efficiency and real situation of the test, the rock model dimensions are 25 mm × 50 mm with 11,979 particles. The microscopic parameters are consistent with Table 1. Rigid particles with a density of 7850 kg/m3 and radius of 6 mm are generated on the right side of the central axis of the rock model as Cr–Ni alloy particles. The particle-impact numerical simulation is conducted by changing the rock confining pressure and applying velocity to the alloy particle.
3.3 Characteristics and formation mechanism of cracks
When the tensile stress or shear stress of the particles inside the rock exceeds its strength limit, the bond between the particles breaks, forming cracks. The change in the rock stress state is a direct cause of rock failure. To analyze the formation mechanism and propagation characteristics of cracks under particle impact, the measurement region is uniformly distributed on the discrete-element model of granite to obtain the stress cloud diagram of the rock and analyze the stress evolution of the rock during the particle impact. The X-direction principal stress and fracture evolution during the impact process under a confining pressure of 0 MPa and particle velocity of 50 m/s are shown in Fig. 15.
After the high-speed alloy particles impact the rock, a compressive stress is generated at the impact point and propagates through the rock as spherical waves. With the decrease in the particle velocity, energy is gradually transferred to the rock in the form of a stress wave, and the compressive stress of the rock near the impact point increases rapidly. The normal and tangential stresses between rock particles increase rapidly under the compressive stress. When the normal and tangential stresses between the particles exceed the normal or tangential bond strengths of the particles, the bonds break, and tensile and shear cracks form. In the vicinity of the impact point, the higher compressive stress causes tensile and shear cracks between the rock particles, and the crack density is higher, which forms a fracture zone. As the fracture zone continues to increase, the energy of the impact stress wave consumed by the formation of new cracks gradually increases. When the energy of the stress wave is not sufficient to break the bond between the rock particles at the boundary of the fracture zone, the boundary of the fracture zone stops extending to the inside of the rock, and only the bond between the crystals with lower strengths breaks. A stress concentration zone is formed at the particle bonding at the end of the crack, and the crack further develops to form an intergranular main crack propagation zone.
The macroscopic failure of rock is the result of the formation and interconnection of micro-cracks caused by the failure of cementation between micro-rock particles under stress. To further analyze the formation and propagation mechanism of microcrack between rock particles under particle impact, two measurement regions are arranged at an interval of 3 mm on the central axis of the particle-impact point, and a stress–time history curve is extracted. As shown in Fig. 16, measurement region A is located in the fracture zone, while measurement region B is located in the intergranular main crack propagation zone.
Figure 17 shows the time history curves of the normal stress in the X direction (σx), normal stress in the Y direction (σy), and shear stress (σxy) in region A and crack number. After 3.5 µs of particle-impact on the rock, the stress wave reaches region A, and thus the normal stress in region A increases rapidly to 94.1 MPa (compressive stress). The shear stress is increased to 25.2 MPa at 12 µs due to the compressive stress. The bonds between a large number of rock particles near region A break down under the shear and tensile stresses, and the number of cracks rapidly increases to 1500. The large number of bond failures leads to energy consumption of the stress wave, the compressive and shear stresses in region A decrease rapidly, and the stress unloading is obvious. During 20–40 µs, the alloy particles compress the broken rock particles in region A, and the compressive stress and maximum shear stress between particles increase again. After 40 µs, the velocity of alloy particles gradually decreased, the rock particles could not be squeezed continuously, and the stress gradually decreased to 0 MPa.
The mechanism of fracture formation and propagation in the fracture zone is shown in Fig. 18. Particle b squeezes particles d and e under the stress wave. Owing to the inertia and Poisson effects, particles d and e move to either side of the tangential direction of the stress wave, resulting in a tensile crack between particles d and e under the tensile stress in the Y direction. Particle b then moves toward the normal direction of the stress wave under the compressive stress, such that the bonds between particles b and a (b–a bond) and b and c (b–c bond) are destroyed under the shear stress, forming shear cracks. The stress wave further acts on particle b, and thus it twists with particles d and e. The b–e and b–d bonds then break under the shear stress to form shear cracks. Therefore, the compressive stress with a large energy is the main reason for the formation of the fracture zone. The compressive stress leads to formation of shear and tensile cracks between rock particles in this region under shear and tensile stresses. The cracks are interconnected to form the fracture zone with a larger degree of fragmentation.
Figure 19 shows the time history curves of the normal stress in the X direction (σx), normal stress in the Y direction (σy), and shear stress (σxy) in region B. The stress wave reaches region B at 4.4 µs and reaches the peak stress of 60.6 MPa (compressive stress) at 34.9 µs, which is considerably smaller than the peak stress of 134 MPa in region A. Different from region A, due to the inertia and Poisson effect, the peak value of the normal stress in the Y direction (Y-direction derived tensile stress) in region B reaches 16 MPa, while the shear stress remains at a low level. The Y-direction derived tensile stress between rock particles exceeds the tensile strength, and the bonds between rock particles break, forming microcracks. The compressive stress squeezes the microcracks to form a stress concentration zone at the end of the crack, and the crack further extends along the intergranular bonds with a lower strength to form the intergranular main crack propagation zone. The mechanism of fracture formation and propagation in the intergranular main crack propagation zone is shown in Fig. 20. The d–e bond is an intergranular bond with a low strength. Particle b squeezes particles d and e under the compressive stress so that the d–e bond exhibits a tensile stress state. When the tensile stress exceeds the tensile strength of the d–e bond, the bond is broken, and an intergranular crack is formed. The crack further expands along the crystal, forming the intergranular main crack propagation zone under the stress wave.
3.4 Effect of confining pressure on crack propagation behavior
Figure 21 shows the distribution of rock fractures impacted by particles under different confining pressures. Percentages of different cracks are shown in Fig. 22. The increase in the confining pressure can effectively suppress the generation of cracks. In the intergranular main crack propagation zone, when the confining pressure increases to 10 MPa, the proportion of intergranular tensile cracks largely decreases, and the proportion of intragranular tensile cracks and intragranular shear cracks increases. When the confining pressure increases to 20 MPa, the proportion of intragranular tensile cracks decreases, and the proportion of intragranular shear cracks increases.
Time history curves of the normal stress in the X direction, normal stress in the Y direction, and shear stress in regions A and B under different confining pressures were extracted to analyze the influence of the confining pressure on the rock-breaking mechanism. Figure 23 shows the stress–time curves of region A under different confining pressures. The initial Y-direction normal stress (σy) and shear stress (σxy) of the rock are 0 MPa when the confining pressure is 0 MPa. After the alloy particles impact the rock, the compressive stress in the X direction increases rapidly, and the shear stress increases accordingly. When the shear stress is increased to 16.3 MPa, the crack extends into region A. The rapid increases in the compressive and shear stresses lead to shear and tension failures among the bonds between rock particles in region A. When the confining pressure is 10 and 20 MPa, the shear stress is 20.6 and 23.6 MPa, respectively, when the crack extends to region A. Compared to the condition without a confining pressure, the shear stress increased by 4.3 and 7.3 MPa, respectively. The increase in the confining pressure leads to an increase in the shear stress required for bond failure between rock particles.
Figure 24 shows the mechanism of fracture formation and propagation in the fracture zone under a confining pressure. After the application of a confining pressure, the initial prestress of the compressive stress is added between particles. Particle b is displaced under the stress wave, and thus it twists against particles a and c, and the b–a and b–c bonds develop a shear stress. When the shear stress reaches the shear strength of the bond, the bond breaks to form shear cracks. As the stress continues to act on the d–e bond, the bond is subjected to tensile and shear stresses. Under the action of the confining pressure, the bond needs to overcome the initial prestress to form a tensile crack. As a result, the d–e bond is more susceptible to form a shear crack under the shear stress. Particle b is then further influenced by the stress wave and thus twists with particles d and e. The b–d and b–e bonds break under the shear stress to form a shear fracture. Compared to the condition without a confining pressure, the cementation between d–e is transformed from a tensile failure to a shear failure.
The increase in the confining pressure inhibits the generation of tensile cracks, which promotes the generation of shear cracks and increases the proportion of shear cracks. As the shear strength of the rock is generally larger than the tensile strength, a higher impact stress wave energy is needed to destroy the rock under a confining pressure. On the other hand, rock particles need to overcome the tangential friction to achieve a shear failure strength to form shear cracks. With the increase in the confining pressure, the particles squeeze each other, which enhances the friction effect between particles, thus increasing the shear stress required to form a shear failure. The increase in the prestress between particles, increase in the shear fracture ratio, and enhancement in the friction effect lead to an increase in the energy required to produce the same number of cracks.
Figure 25 shows the stress–time curves of region B under different confining pressures. When the stress wave propagates to region B, the compressive stress in the X direction increases, resulting in a derived tensile stress in the Y direction. Under a confining pressure of 0 MPa, the derived tensile stress in the Y direction increases to 15.1 MPa. The lower-strength intergranular bonds break down under the derived tensile stress, forming the intergranular main crack propagation zone. When the confining pressure is 10 MPa, a part of the energy of the Y-direction derived tensile stress is used to overcome the prestress between particles, so that the peak value of the tensile stress is only 8.6 MPa, and the cracks in the intergranular main crack propagation area are reduced. When the confining pressure is 20 MPa, the intergranular main crack propagation zone is not formed in region B, as the derived tensile stress has not overcome the prestress to form a tensile stress, and the crack has not extended further to form the intergranular main crack propagation zone.
Figure 26 shows the mechanism of fracture formation and propagation in the intergranular main crack propagation zone under a confining pressure. The stress wave acts on particle b so that the b–e and b–c bonds are subjected to a shear stress, while the d–e bond is subjected to a tensile stress. As the intergranular bond strength is considerably smaller than the intragranular bond strength, the tensile strength of the rock is smaller than the shear strength, and the energy consumption of the stress wave leads to a reduction in the peak value of the shear and tensile stresses between particles. It can only cause a tensile failure of intergranular d–e bond. Particle b continues to squeeze the cracks between d and e, forming a tensile stress concentration zone at the crack end. The cracks continue to expand between crystals to form the intergranular main crack propagation zone. With the increase in the confining pressure, the particles always maintain the compressive stress state, effectively reducing the peak value of the derived tensile stress in the Y direction between d and e and reducing the range of the intergranular main crack propagation zone.
In summary, the confining pressure yielded a prestress between rock particles, so that the derived tensile stress needs to consume energy to overcome the initial compressive stress between particles to form tensile cracks, inhibiting the generation of tensile cracks and intergranular main crack. Simultaneously, the increase in the confining pressure leads to increases in the shear fracture ratio and friction effect between rock particles, which leads to an increase in the energy consumption of crack expansion.